I'm working my way through Ivey and Landsberg, Cartan for Beginners, and I'm working on 2.6.13.2(b). After defining, for $X \in \Gamma(TM)$ a vector field, with section $s:M \rightarrow \mathcal{F}_{ON}(M)$ a (local) orthonormal coframing, we have $X=X^ie_i$ for some functions $X^i$. Define the covariant derivative of $X$ to be $$\nabla X = (dX^i + X^j\eta^i_j)\otimes e_i \in \Omega^1(M,TM) = \Gamma(TM\otimes T*M),$$ with $\eta^i$ and $\eta^i_j$ being pulled back via $s$.
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(b) For $Y \in \Gamma(TM),$ we define $\nabla_YX := Y\neg\nabla X = (dX^i + X^j\eta^i_j)(Y)e_i.$ Show that $\nabla_Y(fX) = f\nabla_YX + X(f)Y$ for $f\in C^{\infty}(M)$....
Here is my question: I've seen multiple explanations (self-learner) of the connection forms defined here as tautological 1-forms on $T\mathcal{F}_{ON}$. I've seen multiple definitions of the connection, Cartan, Koszul, Ehresmann, and one of them which defined it in terms of a splitting of an exact sequence involving the vertical bundle over $\mathcal{F}$ also said that for adapted coordinates, $\eta^i_j(Y)$ becomes $\Gamma^i_{kj}Y^k$.
What is the relationship between the different definitions, and does that latter work for all three different definitions of the connection (one definition, Koszul in Darling, had created $\Gamma(M)$ as a local vector valued form that pulls back from $\mathbb{R}^n$ as well, so it seems like it can be defined generally and consistently enough but I'm very unsure).
I do remember with, say, integrals, the increasingly abstract definitions (Riemann, Stiltjes, Lesbesgue) were sort of nested so that the less general definition didn't change value when one moved to increasing generality, but I've seen no such explanation for all these characterizations of connections.